What is the greatest common factor of $20a^{2}$, $12a^{2}$, and $8a$ ?
Solution: Let's factor each monomial to its prime factors: $\begin{aligned} 20a^{2}&=(2)(2)(5)(a)(a) \\\\ 12a^{2}&=(2)(2)(3)(a)(a) \\\\ 8a&=(2)(2)(2)(a) \end{aligned}$ We want the largest set of factors that's included in all three monomials. All of the monomials have two factors of $ 2$ and one factor of $ a$ : $\begin{aligned} 20a^{2}&=( 2)( 2)(5)( a)(a) \\\\ 12a^{2}&=( 2)( 2)(3)( a)(a) \\\\ 8a&=( 2)( 2)(2)( a) \end{aligned}$ This is the greatest common factor: $( 2)( 2)( a)=4a$